Mohinder S. Grewal - Global Navigation Satellite Systems, Inertial Navigation, and Integration

Here, “CEP” stands for “ circular error probable ” or “ circle of equal probability ,” specified as the radius of the circle about the navigation solution with a radius such that the true solution is equally likely to be inside or outside of that circle. That radius is usually specified in units of nautical miles (1852 m in SI units 6 ).

“Free” here means unaided by external sensors – except for altimeter aiding required to stabilize altitude errors.

Unaided inertial navigators are essentially integrators, integrating sensed accelerations (including sensor noise) to get velocity and integrating velocity to get position. Unfortunately, integration does bad things to zero‐mean random noise. The integral of zero‐mean additive uncorrelated random noise on an accelerometer output is a random walk process, the variance of which grows linearly over time. Integrated twice to get position, we might expect variance to grow quadratically with time, in which case its standard deviation would grow linearly with time. As a consequence, one might expect INS performance to be characterized by how fast an error distance statistic grows linearly with time.

A common INS performance metric used in military applications is CEP rate , generally calculated over typical INS mission times (a few hours, generally) and originally measured (before GNSS availability) during INS performance qualification trials aboard different military aircraft or surface vehicles on suitably instrumented test ranges. That can now be done using GNSS as the independent position solution for CEP rate determinations.

In the 1970s, before GPS became a reality, the US Department of Defense established the following categories of INS performance:

High accuracy systems have free inertial CEP rates in the order of 0.1 nautical miles per hour ( m/h) or better.

Medium accuracy systems have free inertial CEP rates in the order of 1 nautical mile per hour ( km/h). This was the level of accuracy deemed sufficient for most military and commercial aircraft [11].

Low accuracy systems have free inertial CEP rates in the order of 10 nautical miles per hour ( km/h) or worse. Sometimes called tactical grade INS performance, this range covered requirements for many short‐range standoff weapons such as guided artillery or tactical missiles.

Order‐of‐magnitude ranges for the inertial sensor errors in these INS performance categories are summarized in Table 3.1. Inertial sensors below tactical grade are sometimes called commercial grade or consumer grade .

Table 3.1INS and inertial sensor performance ranges.

	Performance ranges
System or sensor	High	Medium	Low	Units
INS				NMi/h CEP rate
Gyroscopes				deg/h drift rate
Accelerometers				(9.8 m/ ) bias

Unfortunately, the probability‐based CEP statistic is not exactly compatible with the RMS statistics used in the Kalman filter Riccati equations 7 for characterizing INS performance. One must assume some standard probability density function (e.g. Gaussian) to convert mean‐squared horizontal position errors to CEPs.

1 Inertial navigation accuracy is mostly limited by inertial sensor accuracy.

2 The accuracy requirements for inertial sensors cannot always be met within manufacturing tolerances. Some form of calibration is usually required for compensating the residual errors.

3 INS accuracy degrades over time, and the most accurate systems generally have shortest mission times. For example, ICBMs only need their inertial systems for a few minutes.

4 Performance of inertial systems is commonly specified in terms of CEP rate.

5 Accelerometers cannot measure gravitational acceleration.

6 Both inertial and satellite navigation require accurate models of the Earth's gravitational field.

7 Both navigation modes also require an accurate model of the shape of the Earth.

8 The first successful navigation systems were gimbaled, in part because the computer technology required for strapdown implementations was decades away. That has not been a problem for about four decades.

9 Gimbaled systems tend to be more accurate and more expensive than strapdown systems.

10 The more reliable attitude implementations for strapdown systems use quaternions to represent attitude.

11 Systems traditionally go through a testing and evaluation process to verify performance.

12 Before testing and evaluation of an INS, its expected performance is commonly evaluated using the analytical models of Chapter 11.

Inertial navigation has a rich and growing technology base – more than can be covered in a single book, and certainly not in one chapter – but there is some good open‐source literature on the subject:

1 Titterton and Weston [12] is a good source for additional information on strapdown hardware and software.

2 Paul Savage's two volume tome [13] on strapdown system implementations is also rather thorough.

3 Chapter 5of [14] and the references therein include some recent developments.

4 Journals of the IEEE, IEE, Institute of Navigation, and other professional engineering societies generally have the latest developments on inertial sensors and systems.